You could probably work a silver coin without heating it but it would just be more difficult and time consuming. Any coin that is comprised mostly of nickel (which most modern coins are) would be an absolute bastard to hammer into a ring without heat.

Interesting. Thank you for taking the time to reply! I do appreciate the tips, especially from someone putting out such nice end results.

Last question that I don't think I've seen answered anywhere: how do you judge how large to make the initial hole? Is it a fixed size per the diameter of the coin, or is there an equation where hole size relates to possible ring size in the end?

You're welcome, I love talking about this stuff. I decide what size hole to punch in the coin based on the intended size of ring and how much of the coins details I want to retain. Sometimes people want a ring with a more narrow band so I would punch a bigger hole. If they want a wide band then I'd punch a smaller hole. Generally I am going for a band width of about 7 - 9 mm as that gives a fairly balanced ring that isn't too wide or narrow.

Gotcha. So the hole size is mostly for ring width, and I assume that ring size is more limited by coin size and the amount of "stretching" you get in the end result (if you have to widen the ring much past the original diameter).